Summary: Part II
Geometry
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If you have not yet read the preface, then please do so now.
Since you have read the preface, you already know that central to much of what
we shall be looking at in Part III is the geometry of excursion sets:
Au Au(f, T ) = {t T : f (t) u} f -1
([u, ))
for random fields f over general parameter spaces T .
In order to do this, we are going to need quite a bit of geometry. In fact, we are
going to need a lot more than one might expect, since the answers to the relatively
simple questions that we shall be asking end up involving concepts that do not, at
first sight, seem to have much to do with the original questions.
In Part III we shall see that an extremely convenient description of the geometric
properties of A is its Euler, or Euler­Poincaré, characteristic. In many cases, the
Euler characteristic is determined by the fact that it is the unique integer-valued
functional , defined on collections of nice A, satisfying
(A) =
0 if A = ,
1 if A is ball-like,